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Properties

Label 840.2.w.b
Level $840$
Weight $2$
Character orbit 840.w
Analytic conductor $6.707$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [840,2,Mod(139,840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("840.139"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.w (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{3} + 48 q^{9} + 8 q^{10} - 2 q^{14} + 8 q^{16} + 4 q^{20} + 48 q^{27} + 14 q^{28} + 8 q^{30} + 8 q^{35} - 12 q^{38} + 8 q^{40} - 2 q^{42} + 4 q^{44} - 8 q^{46} + 8 q^{48} - 12 q^{50} - 36 q^{52}+ \cdots + 40 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1 −1.40472 − 0.163597i 1.00000 1.94647 + 0.459616i 1.79945 + 1.32740i −1.40472 − 0.163597i 1.34072 + 2.28089i −2.65905 − 0.964069i 1.00000 −2.31056 − 2.15901i
139.2 −1.40472 + 0.163597i 1.00000 1.94647 − 0.459616i 1.79945 − 1.32740i −1.40472 + 0.163597i 1.34072 − 2.28089i −2.65905 + 0.964069i 1.00000 −2.31056 + 2.15901i
139.3 −1.38753 − 0.273429i 1.00000 1.85047 + 0.758780i −0.840958 − 2.07190i −1.38753 − 0.273429i −2.31981 + 1.27219i −2.36011 − 1.55880i 1.00000 0.600336 + 3.10477i
139.4 −1.38753 + 0.273429i 1.00000 1.85047 − 0.758780i −0.840958 + 2.07190i −1.38753 + 0.273429i −2.31981 − 1.27219i −2.36011 + 1.55880i 1.00000 0.600336 − 3.10477i
139.5 −1.34616 − 0.433405i 1.00000 1.62432 + 1.16687i −2.16047 + 0.576496i −1.34616 − 0.433405i 0.660553 − 2.56197i −1.68087 − 2.27479i 1.00000 3.15821 + 0.160302i
139.6 −1.34616 + 0.433405i 1.00000 1.62432 − 1.16687i −2.16047 − 0.576496i −1.34616 + 0.433405i 0.660553 + 2.56197i −1.68087 + 2.27479i 1.00000 3.15821 − 0.160302i
139.7 −1.31790 − 0.512983i 1.00000 1.47370 + 1.35212i −1.36576 + 1.77051i −1.31790 − 0.512983i 2.64396 − 0.0973159i −1.24857 − 2.53793i 1.00000 2.70817 − 1.63274i
139.8 −1.31790 + 0.512983i 1.00000 1.47370 − 1.35212i −1.36576 − 1.77051i −1.31790 + 0.512983i 2.64396 + 0.0973159i −1.24857 + 2.53793i 1.00000 2.70817 + 1.63274i
139.9 −1.21556 − 0.722780i 1.00000 0.955179 + 1.75717i 2.09340 − 0.785912i −1.21556 − 0.722780i −2.63147 + 0.274565i 0.108965 − 2.82633i 1.00000 −3.11270 − 0.557746i
139.10 −1.21556 + 0.722780i 1.00000 0.955179 − 1.75717i 2.09340 + 0.785912i −1.21556 + 0.722780i −2.63147 − 0.274565i 0.108965 + 2.82633i 1.00000 −3.11270 + 0.557746i
139.11 −0.965584 − 1.03327i 1.00000 −0.135296 + 1.99542i −0.836142 − 2.07385i −0.965584 − 1.03327i 0.292578 − 2.62952i 2.19245 − 1.78695i 1.00000 −1.33549 + 2.86644i
139.12 −0.965584 + 1.03327i 1.00000 −0.135296 − 1.99542i −0.836142 + 2.07385i −0.965584 + 1.03327i 0.292578 + 2.62952i 2.19245 + 1.78695i 1.00000 −1.33549 − 2.86644i
139.13 −0.911149 − 1.08158i 1.00000 −0.339615 + 1.97095i −2.20852 − 0.349934i −0.911149 − 1.08158i 0.795986 + 2.52317i 2.44118 − 1.42851i 1.00000 1.63381 + 2.70752i
139.14 −0.911149 + 1.08158i 1.00000 −0.339615 − 1.97095i −2.20852 + 0.349934i −0.911149 + 1.08158i 0.795986 − 2.52317i 2.44118 + 1.42851i 1.00000 1.63381 − 2.70752i
139.15 −0.752597 − 1.19733i 1.00000 −0.867194 + 1.80221i 1.64690 + 1.51252i −0.752597 − 1.19733i 2.63299 − 0.259532i 2.81049 − 0.318025i 1.00000 0.571535 − 3.11020i
139.16 −0.752597 + 1.19733i 1.00000 −0.867194 − 1.80221i 1.64690 − 1.51252i −0.752597 + 1.19733i 2.63299 + 0.259532i 2.81049 + 0.318025i 1.00000 0.571535 + 3.11020i
139.17 −0.716255 − 1.21942i 1.00000 −0.973958 + 1.74683i 0.151655 + 2.23092i −0.716255 − 1.21942i −2.00273 − 1.72889i 2.82771 − 0.0635119i 1.00000 2.61180 − 1.78284i
139.18 −0.716255 + 1.21942i 1.00000 −0.973958 − 1.74683i 0.151655 − 2.23092i −0.716255 + 1.21942i −2.00273 + 1.72889i 2.82771 + 0.0635119i 1.00000 2.61180 + 1.78284i
139.19 −0.355930 − 1.36869i 1.00000 −1.74663 + 0.974317i 1.14400 − 1.92127i −0.355930 − 1.36869i −1.89556 − 1.84576i 1.95522 + 2.04380i 1.00000 −3.03680 − 0.881940i
139.20 −0.355930 + 1.36869i 1.00000 −1.74663 − 0.974317i 1.14400 + 1.92127i −0.355930 + 1.36869i −1.89556 + 1.84576i 1.95522 − 2.04380i 1.00000 −3.03680 + 0.881940i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.48
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
35.c odd 2 1 inner
280.n even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.w.b yes 48
4.b odd 2 1 3360.2.w.a 48
5.b even 2 1 840.2.w.a ✓ 48
7.b odd 2 1 840.2.w.a ✓ 48
8.b even 2 1 3360.2.w.a 48
8.d odd 2 1 inner 840.2.w.b yes 48
20.d odd 2 1 3360.2.w.b 48
28.d even 2 1 3360.2.w.b 48
35.c odd 2 1 inner 840.2.w.b yes 48
40.e odd 2 1 840.2.w.a ✓ 48
40.f even 2 1 3360.2.w.b 48
56.e even 2 1 840.2.w.a ✓ 48
56.h odd 2 1 3360.2.w.b 48
140.c even 2 1 3360.2.w.a 48
280.c odd 2 1 3360.2.w.a 48
280.n even 2 1 inner 840.2.w.b yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.w.a ✓ 48 5.b even 2 1
840.2.w.a ✓ 48 7.b odd 2 1
840.2.w.a ✓ 48 40.e odd 2 1
840.2.w.a ✓ 48 56.e even 2 1
840.2.w.b yes 48 1.a even 1 1 trivial
840.2.w.b yes 48 8.d odd 2 1 inner
840.2.w.b yes 48 35.c odd 2 1 inner
840.2.w.b yes 48 280.n even 2 1 inner
3360.2.w.a 48 4.b odd 2 1
3360.2.w.a 48 8.b even 2 1
3360.2.w.a 48 140.c even 2 1
3360.2.w.a 48 280.c odd 2 1
3360.2.w.b 48 20.d odd 2 1
3360.2.w.b 48 28.d even 2 1
3360.2.w.b 48 40.f even 2 1
3360.2.w.b 48 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{12} - 108 T_{17}^{10} - 4 T_{17}^{9} + 4008 T_{17}^{8} + 64 T_{17}^{7} - 62112 T_{17}^{6} + \cdots + 65536 \) acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\). Copy content Toggle raw display